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kent davidge
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I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?
Can you give us a wild such case?WWGD said:This is true only if V is assumed continuous, otherwise its values can vary pretty wildly even in small regions.
Sure, let me think a bit. If you want just a scalar field, consider any subset S of the interior and use its characteristic function, making sure S has measure 0, e.g., for scalar fields, the Char function of ( a finite collection of) Rationals. The integral will be 0 ( and will exist for a finite collection) but the function is not identically zero.MathematicalPhysicist said:Can you give us a wild such case?
I am not saying you're not correct, I am just not that good in finding examples.
No, not at all.kent davidge said:I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?
No. This is the other way around. A constant vector field is conservative. You must not conclude from free of rotations to vanishing!MathematicalPhysicist said:Yes because you can take your enclosed area to be small enough such that ##\vec{V}## is constant in it.
Even if! My example is simply connected and the area closed if you choose a disc around zero; or I misunderstood the condition. Stokes does not mean that the integrand is zero! If you run around in an electrical potential, you will not gain or lose energy. That doesn't mean there is no field.WWGD said:Dont we need simple-connectedness and a simple-closed curve? Maybe this is assumed?
The integral of a vector field is a mathematical concept that represents the total magnitude and direction of a vector field over a given region. It is essentially a way to measure the total amount of a vector quantity, such as force or velocity, within a specific area.
The integral of a vector field is calculated by breaking down the vector field into infinitesimally small pieces, known as infinitesimal line elements. These elements are then multiplied by the magnitude of the vector at that point, and integrated over the given region. The result is a single value that represents the total amount of the vector field within that region.
In physics, the integral of a vector field is often used to calculate the work done by a force or the flux of a vector field through a surface. It is also used in the study of fluid dynamics to calculate the flow rate of a fluid through a given area. In general, the integral of a vector field is a useful tool for understanding and analyzing physical phenomena.
Yes, the integral of a vector field can be negative. This occurs when the vector field is pointing in a direction opposite to the direction of integration. In this case, the vector field is considered to have a negative contribution to the total integral value.
A line integral is the integral of a vector field along a one-dimensional curve, while a surface integral is the integral of a vector field over a two-dimensional surface. In other words, a line integral represents the total amount of a vector field along a specific path, while a surface integral represents the total amount of a vector field over a given area.